Optimal Strouhal number for swimming animals

Christophe Eloy

Department of Mechanical and Aerospace Engineering, University of California San Diego, La Jolla, USA

IRPHE, CNRS & Aix-Marseille University, Marseille, France

Abstract

To evaluate the swimming performances of aquatic animals, an important dimensionless quantity is the Strouhalnumber, St = f A/U, with f the tail-beat frequency, A the peak-to-peak tail amplitude, and U the swimming velocity.Experiments with flapping foils have exhibited maximum propulsive efficiency in the interval 0.25 < St < 0.35 andit has been argued that animals likely evolved to swim in the same narrow interval. Using Lighthill’s elongated-bodytheory to address undulatory propulsion, it is demonstrated here that the optimal Strouhal number increases from 0.15to 0.8 for animals spanning from the largest cetaceans to the smallest tadpoles. To assess the validity of this model, theswimming kinematics of 53 different species of aquatic animals have been compiled from the literature and it showsthat their Strouhal numbers are consistently near the predicted optimum.

Keywords: Swimming, Strouhal number, Constrained optimisation, Potential flow theory

1. Introduction

1.1. Strouhal numberLord Rayleigh (1915) was the first to define the Strouhal (1878) number, St, to quantify in a proper dimensionless

fashion the frequency of vortex shedding behind a bluff body. A decade later, this definition was eventually changedby Benard (1926) to be the inverse of Rayleigh’s suggestion: St = f d/U, where f is the frequency, d is the diameterof the bluff body and U is the flow velocity.

The Strouhal number is intimately linked to the arrangements of vortices in the wake as already pointed out byRayleigh (1915). Von Karman (1911) showed that two infinite rows of point vortices are always unstable unless theirspacing ratio has a particular value b/a = 0.281 (see Fig. 1a). Assuming that the vortices in the wake travel at thevelocity Uw < U, the vortex shedding frequency is then f = Uw/a and the Strouhal number is linked to the spacingratio through St = (b/a)(d/b)(Uw/U). The Strouhal number can therefore be predicted based on estimation of thespreading factor b/d and the velocity ratio Uw/U (Roshko, 1954).

A more modern approach to predict the Strouhal number consists in analysing the local stability properties of thewake, a method reviewed by Huerre and Monkewitz (1990). To do so, a base flow is considered which can eitherbe a steady solution of the Navier–Stokes equations around the bluff body or the time-average flow obtained throughexperiments or numerical simulations. Pier (2002) has shown that, in the near wake of this base flow, a transition fromconvective to absolute instability occurs. This region acts as a source generating disturbances advected and amplifieddownstream and tunes the entire wake to its frequency, thus selecting the Strouhal number.

In the context of swimming, the Strouhal number has been introduced in the nineties by Triantafyllou et al. (1991,1993) with two innovative papers (see also some recent reviews on swimming: Sfakiotakis et al., 1999; Triantafyllouet al., 2000; Lauder and Tytell, 2005). It is defined as

St =f AU, (1)

Email address: christophe.eloy@irphe.univ-mrs.fr (Christophe Eloy)

Preprint submitted to Journal of Fluids and Structures December 10, 2011

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FIG. 2. Color Instantaneous spanwise vorticity fields left column, s!1 and mean flow time averaged horizontal velocity, right column,

m s!1 for fixed Strouhal and Reynolds numbers Sr=0.22 and Re=255 and from top to bottom, for AD=0.36,0.71,1.07,1.77, and 2.8. The

field of view placed at midheight of the foil covers from !2D to 20D on the horizontal streamwise direction x and !8D to 8D in the

vertical crossstream direction y where the origin is defined at the trailing edge of the flap at zero angle of attack.

TRANSITIONS IN THE WAKE OF A FLAPPING FOIL PHYSICAL REVIEW E 77, 016308 2008

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FIG. 2. Color Instantaneous spanwise vorticity fields left column, s!1 and mean flow time averaged horizontal velocity, right column,

m s!1 for fixed Strouhal and Reynolds numbers Sr=0.22 and Re=255 and from top to bottom, for AD=0.36,0.71,1.07,1.77, and 2.8. The

field of view placed at midheight of the foil covers from !2D to 20D on the horizontal streamwise direction x and !8D to 8D in the

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TRANSITIONS IN THE WAKE OF A FLAPPING FOIL PHYSICAL REVIEW E 77, 016308 2008

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Figure 1: Schematic view of the (a) Benard-von Karman (BvK) vortex street behind a circular cylinder and (b) reverse Benard-von Karman (rBvK)vortex street behind a swimming fish. The lines in the wakes illustrate what can be obtained typically with dye visualisations. The averageperturbation flow u(y) in the far wake is a jet toward the cylinder (c) and away from the fish (d) respectively. Both of these jets are surrounded by aregion of counterflow.

where f is the tail-beat frequency, A is the peak-to-peak amplitude at the tail tip and U is the average swimming speed.The argument of Triantafyllou et al. (1993, 1991) relies on the observation that the wake behind a swimming animalresembles the Benard-von Karman (BvK) vortex street observed behind bluff bodies except that the sign of vorticesare inverted giving a reverse Benard-von Karman (rBvK) street (see Fig. 1b).

In the BvK street, the average flow exhibits a deficit of velocity compared to the imposed flow U, indicating thatlongitudinal momentum has been lost and that a drag force is exerted on the bluff body (Fig. 1c). However, swimminganimals are self-propelled and therefore no net drag nor thrust is exerted on average when they swim at constant speed:the resulting rBvK wake is therefore momentumless and exhibits on average a jet around the centerline surrounded bya region of counterflow (Fig. 1d).

Applying similar techniques to the ones used to study the stability of bluff body wakes, Triantafyllou et al. (1991,1993) have shown that wakes associated to net thrust are only convectively unstable (there is no region of absoluteinstability). Such wakes acts as amplifier in a narrow range of frequencies which was found to correspond to theinterval 0.25 < St < 0.35, for a family of two-dimensional wakes obtained by fitting the experimental results ofKoochesfahani (1989). They argued that swimming animals likely evolved to exploit this amplification to reduce theswimming costs and hence should be observed to swim in the same narrow interval of Strouhal numbers. In parallel,experiments have been carried out by the same group (Triantafyllou et al., 1993; Anderson et al., 1998; Read et al.,2003; Schouveiler et al., 2005) with rigid airfoils submitted to harmonic flapping, confirming that maximum efficiencycould be reached in the same interval.

In their papers, Triantafyllou et al. (1991, 1993) analysed twelve species (dolphins, sharks, some scombroidsand other bony fishes) whose swimming kinematics were found in the literature and concluded that most of theseswimming animals indeed swim in the interval 0.25 < St < 0.35. More recently, Taylor et al. (2003) have shownthat birds, bats and insects in cruising flight flap their wings within a similar narrow range of Strouhal number,0.2 < St < 0.4. In contrast with this apparent universal range, it has been observed (with no physical explanation) thatthe Strouhal number decreases as a fish swim faster (Fig. 11.3B in Lauder and Tytell, 2005) or as different species oflarger size are considered (Kayan et al., 1978).

In this work, an optimal Strouhal number will be calculated for swimming animals without explicit reference tothe stability or the dynamics of their wakes. First, Lighthill’s large-amplitude elongated-body theory will be intro-duced and discussed. In section 2 the optimal motion of the animal tails will be calculated by solving a constrainedminimisation problem. In section 3, the predicted optimal Strouhal number will be compared to the observations ondifferent species of swimming animals found in the literature. Finally, these results will be discussed in section 4.

1.2. Lighthill’s elongated-body theory

Consider an aquatic animal of length L performing undulatory propulsion with constant mean velocity U in thex-direction (see Fig. 2). The position of its body at time t is described by the position (x, y) of any point of the

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backbone. The plane Oxy would be the horizontal plane for fishes and the vertical plane for cetaceans. Defining thecurvilinear coordinate s as the distance from the tail tip, the functions x(s, t) and y(s, t) fully describe the kinematicsof swimming. The velocity of any point on the backbone is the time-derivative of the position, v = (x, y), which canbe decomposed into tangential and normal velocities (see Fig. 2)

u = xx′ + yy′, (2a)w = yx′ − xy′, (2b)

where the primes and dots denote differentiation with respect to s and t respectively.Elongated-body theory makes use of the small aspect ratio of the swimming animal. When h � L, the forces

acting on each cross section can be assumed to be the same as those acting on an infinite cylinder with same crosssection and moving with the same velocity (u,w), even if Candelier et al. (2011) have shown that, for motions of largeamplitude, this is not strictly true. The main idea behind Lighthill’s elongated-body theory (Lighthill, 1971) is thento treat perpendicular motions (given by the velocity w) reactively and the tangential motions (given by u) resistively.The elongated-body approximation is therefore valid if the animal is elongated enough such that h � L, if the crosssection varies smoothly along the backbone, and if the Reynolds number, defined as

Re = UL/ν, (3)

with ν the kinematic viscosity, is asymptotically large (more discussion on the validity of this theoretical frameworkwill be given below).

The origin of the reactive force is the conservation of momentum. It can be understood if one realises that, as theanimal swims, a certain volume of water has to be accelerated. This means that a certain force has to be applied tothe water and reactively, the opposite force applies to the animal. The reactive force has been calculated by Lighthill(1971) and its remarkable feature is that its time-average depends only on the motion of the tail. Thus the motion ofthe rest of the body does not need to be known. The same holds true for the kinetic energy given to the fluid per unittime which is the only source of power loss in the elongated-body approximation.

Following Lighthill (1971), the mean thrust 〈T 〉 (which is the reactive force on the animal projected on the x-direction) and the power lost in the wake 〈E〉 are given by

〈T 〉 = 〈m[w

(y − 1

2 wx′)]

s=0〉, (4a)

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]s=0〉, (4b)

where the chevrons denote time-average, m = ρπh2/4 is the added mass per unit length at the tail tip (at s = 0) and ρis the density of water.

In the case of steady swimming, the thrust has to compensate the drag D on average such that 〈T 〉 = D. Hence,the role of viscosity being limited to setting the drag, the only relevant parameters are the added mass at the tail tipm, the swimming velocity U and the drag D. Out of these three parameters, a unique dimensionless quantity can beconstructed which measures the ratio between the drag D and the typical thrust mU2. This new dimensionless numberwill be called the Lighthill number in the following

Li =πD

2mU2 =Sh2 Cd, (5)

where S is the total surface of the animal (or wetted surface) and Cd is the drag coefficient such that D = 12ρU2S Cd.

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Another way to introduce the Lighthill number would be to make the problem dimensionless by using L and U ascharacteristic length and velocity respectively. The dimensionless form of the equation 〈T 〉 = D would then be

〈[w∗

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2 w∗x∗′)]

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Li, (6)

where x∗ = x/L, y∗ = y/L, s∗ = s/L, w∗ = w/U, t∗ = tU/L are all dimensionless quantities. In (6), it is clear thatthe only relevant parameter to the present problem is Li which gathers informations on the geometry of the swimminganimal (through the ratio S/h2) and on the Reynolds number (through the drag coefficient Cd). The optimal motion ofthe tail will thus be a function of Li alone, as it will be shown below. This has to be contrasted with bluff body wakeswhere the Strouhal number is a function of Re, as it has been shown by Rayleigh (1915).

2. Optimisation

2.1. Constrained optimisation problem

Consider now that the incident angle (i.e. the angle between the tail and the swimming direction) is given in thevicinity of the tail tip by the harmonic function

θ(s, t) = θ0 cos (ωt) , for s � 1. (7)

Here the curvature θ′ has been assumed to be zero at s = 0 as it should be the case if one assumes that the tail is elasticand that the internal torque at the tail tip is zero. Taking the cosine and sine of θ yields x′ and y′, which appear asinfinite sums of even and odd harmonics respectively (formulas 9.1.44–45 in Abramowitz and Stegun, 1965)[

x′]

s=0 = cos θ = J0(θ0) − 2J2(θ0) cos (2ωt) + · · · , (8a)[y′]

s=0 = sin θ = 2J1(θ0) cos (ωt) + · · · , (8b)

where Jν(x) is the Bessel function of the first kind. The higher harmonics will be neglected in the following owing tothe fact that they have a negligible influence on the final result.

To calculate the tangential and normal velocities given by (2a,b), the functions x and y need to be known at thetail. Keeping the same harmonics as in (8a,b), the general form of these functions is

[x]s=0 = U + αU cos(2ωt + φ), (9a)[y]

s=0 = πSt U cos(ωt + ψ), (9b)

where φ and ψ are unknown phases, α is a dimensionless amplitude and St is the Strouhal number given by (1).Inserting (2a,b), (8a,b) and (9a,b) into (4a,b) and calculating the time-averages allows to express the mean thrust

〈T 〉 and the mean power loss 〈E〉 as a function of the five dimensionless variables: θ0, St, α, φ and ψ. The constrainedoptimisation problem then consists in finding

min 〈E〉 such that

〈T 〉 = D,0 ≤ (St, α) < ∞,0 ≤ θ0 ≤ π/2,0 ≤ (φ, ψ) < 2π.

(10)

This problem has been solved for 100 different values of the Lighthill number in the interval 0.01 < Li < 1 using thefunction fmincon in M (The MathWorks, Inc., Natick, MA, USA). The results are a predicted optimal Strouhalnumber St(Li) and optimal angle θ0(Li) which are both monotonically increasing functions of Li (see Figs. 4–5 below).This optimisation also shows that, for any value of the Lighthill number, the optimal set of dimensionless variables isalways such that α = ψ = 0. The functions x and y can thus be written in a simpler form for the optimal cases

[x]s=0 = U, (11a)[y]

s=0 = V[y′]

s=0 , (11b)

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Figure 3: Froude efficiency as a function of the Lighthill number for the optimal case (solid line) and for the acceptable range (dashed line).

where V appears as a wave speed at the tail tip and is given by identifying (11b) with (8b) and (9b) when ψ = 0

V =πSt

2J1(θ0)U. (12)

The wave speed V is always greater than the swimming speed U and the ratio U/V is customarily called the slip ratio.The fact that y and y′ are in phase in the optimal case (i.e. ψ = 0) could have been anticipated since the same

holds true in the linear limit, as shown by Lighthill (1970). The relations (11a,b) mean that a simpler version ofthe optimisation can be performed with only two variables, θ0 and St (or St and U/V alternatively), leading to thesame results. Note also that since x does not depend on time, the path followed by the tail tip in the frame ofreference attached to the animal is a straight line in the y-direction. In other words, the figure of eight observed insome experiments (Gray, 1933; Webber et al., 2001) which exists only if α , 0 is not optimal within the presentelongated-body framework.

To estimate the range on which the Strouhal number can change without affecting appreciably the swimmingperformances, the Froude efficiency η is introduced

η =DU

DU + 〈E〉, (13)

which expresses the ratio between the average useful power 〈TU〉 = DU and the total power spent for swimming. Fora given Lighthill number, the constrained optimisation yields a maximum efficiency ηmax(Li). In the following, anyStrouhal number leading to an efficiency greater than ηmax − 0.1 will be considered as acceptable (Fig. 3).

2.2. Limit of validity

Elongated-body theory is inviscid in nature. As a results, all the viscous effects (i.e. the resistive forces) aregathered into a single drag force D, which is usually not modeled. In fact, it is still a controversial issue today toknow whether this drag force is enhanced or reduced by the swimming motion (Barrett et al., 1999; Anderson et al.,2001). It has even been argued that the very idea of separating thrust and drag is impossible because they balanceeach other on average during steady swimming (Schultz and Webb, 2002; Fish and Lauder, 2006; Shirgaonkar et al.,2009). However, this argument can be disputed: skin friction drag, which is the main source of drag for streamlinedbodies, can always be defined, if not measured. Lighthill (1971), quoting discussions with Bone, proposed what issometimes called the ‘Bone–Lighthill boundary-layer thinning hypothesis’: they suggested that the boundary layermay be ‘compressed’ or ‘thinned’ by the body motion, resulting in larger velocity gradients and thus enhanced skinfriction. Although this effect was evidenced by Anderson et al. (2001) by measuring boundary layer velocity profiles

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on swimming fish, the Bone–Lighthill hypothesis remains to be tested and quantified. Therefore, to compare thepresent optimisation calculation (for which a model for D is not needed) with measurements on aquatic animals, thedrag force D will have to be estimated with an empirical formulation, as explained below.

Another key hypothesis of elongated-body theory is that the resistive forces corresponding to perpendicular mo-tions can be neglected. Assuming that this force acts on the whole length L and that its drag coefficient is of order one(as it is the case for a cylinder), its x-projection scales as

Fresistive ∼ ρhw2Ly′, (14)

and has to be negligible in comparison with the reactive force which scales as

Freactive ∼ mwy. (15)

This is true if (Lw)/(hV) � 1 and since w ∼ U(1 − U/V)St, this corresponds to

UV

(1 −

UV

)St �

hL. (16)

This condition will be fulfilled for most animals with a fairly wide tail. However, elongated fishes such as eelsand lampreys, for which h/L < 0.1, will not in general meet this criterion. For these animals, a model taking intoaccount the resistive normal force would be necessary. This may seem counterintuitive, but it means that Lighthill’selongated-body theory (Lighthill, 1971) is not valid when the body is too elongated.

The last assumption behind elongated-body theory is that the animal cross section varies on a typical length scalelarger than h. This is not true for scombrids, dolphins and sharks that share a large aspect-ratio tail. For these animals,a two-dimensional approach would be more suited to study the propulsive performance of the tail. However, theexisting two-dimensional models (Lighthill, 1970; Wu, 1971) are linear and do not allow the same sort of optimisationcalculation as the one presented here.

It can be difficult to assess the validity of Lighthill’s elongated-body theory because there has not been any faircomparison with numerical results so far. The recent paper by Candelier et al. (2011) is one notable exception in whichthey ran numerical simulations for an eel-like swimmer with aspect ratio h/L = 0.1, slip ratio U/V = 0.4, Reynoldsnumber Re = 6 × 105, and for two different Strouhal numbers St = 0.2 and 0.8. In the first case (St = 0.2), thevalidity condition (16) is fairly satisfied (the left-hand side being equal to 0.048) and the thrust is well approximatedby Lighthill’s elongated-body theory. In the second case (St = 0.8), the condition (16) is not satisfied and viscousforces play a significant role in the thrust. Note that, in these simulations, the swimmers are not self-propelled andmore numerical results are clearly needed to assess the validity of Lighthill’s elongated-body theory, in particular forlarger aspect ratio (h/L ≈ 0.2).

3. Comparison with aquatic animals

To assess the validity and the interest of the present theoretical results, the swimming kinematics of variousspecies of aquatic animals have been compiled. Comparing these experiments and the model relies on the implicitassumption that aquatic animals swim optimally or, equivalently, that that their Froude efficiencies are maximised.This assumption is far from being obvious and one could argue that some species have evolved in ecological nicheswhere economical steady swimming is not crucial. One could also remark that Froude efficiency is only a part of thefull picture: the complete energy cost should take into account the efficiency of muscles (which depends critically onfrequency and amplitude) and the losses due to the viscoelasticity of soft tissues or damping at the intervertebral joints(Cheng et al., 1998; Long, 1992; Long et al., 2002; McMillen and Holmes, 2006).

3.1. Preliminary remarks on experimental studiesThe comparison with aquatic animals has to cope with several limitations of the experimental methods, some of

which are listed below.

1. The measurements should be made when the animal is swimming steadily. Any small positive or negativeacceleration can alter the results because the mean thrust is no longer equal to the drag in that case (Videler andHess, 1984).

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2. The drag on swimming animals has been recognised to be difficult to measure adequately (Anderson et al.,2001; Fish and Rohr, 1999; Wu, 2011). It usually depends on the swimming velocity (Videler, 1981) and canbe greatly increased when the animal is close to the water surface (Videler, 1993).

3. In most experimental works cited here, the Strouhal number have not been calculated directly by the authors.It results that St had to be calculated using the average tail amplitude 〈A〉, the average frequency 〈 f 〉 and theaverage swimming velocity 〈U〉. When the quantities vary over large intervals, it can result in non negligibleerrors since the mean of a product is usually not equal to the product of the means. The same holds true for theLighthill number.

4. Most kinematics studies have been performed in water channels (also called flumes or tunnels) where the ani-mals swim against the current imposed by the experimentalist. The presence of walls close to the animal andthe turbulence rate can affect the swimming mode as it has been shown by Webb (1993). This effect can beparticularly large when the experiments are performed in respirometers where the volume of the test section hasto be small enough to allow correct measurements of the variation of oxygen concentration.

5. The wave speed V usually depends on the curvilinear coordinate and it can be shown to either accelerate towardto tail tip (Gillis, 1997) or decelerate (Videler and Hess, 1984). This wave speed can also be calculated throughthe apparent wavelength λ of the animal deflection as V = λ f , but this method usually gives a result differentfrom the direct measurement (Webb et al., 1984). These differences come from the fact that the wave speedis non local in nature: contrarily to the Strouhal number and the maximum angle it does not only depend onmeasurements made at the tail tip.

The limitations of the experimental studies listed above make the comparison with the present analysis difficult.In particular, the Lighthill number can only be estimated in most cases (if not because of the lack of geometricalmeasurements, because of the drag coefficient). The other important point is that the optimality of swimming cannever be guaranteed.

3.2. MethodsDespite the limitations listed above, most of the data available in the literature on swimming kinematics of aquatic

animals have been compiled. From these sources, the Lighthill and the Strouhal numbers have been determinedtogether with the maximum angle at the tail tip and the slip ratio U/V when possible. The following methods havebeen used to extract the experimental data.

When it was possible, the value chosen for the swimming velocity was 75% of the critical velocity Ucrit. Thecritical velocity, as introduced by Brett (1964), measures the maximum sustained speed for a given time (between2 and 30 minutes depending on the authors and on the species). The reason to choose this particular value of thevelocity is that it allows a large enough swimming speed (for lower speed, the swimming mode may not be optimal)without being too close to the critical value where data is usually lacking. Note that, in most studies, Ucrit has not beenmeasured, and that this somewhat arbitrary choice of 75% of the critical velocity does not qualitatively influence theresults.

To evaluate the tail span of a given species, the following rules have been used. When available, the data found inthe source papers have been used. Otherwise, pictures have been collected on Internet and used to estimate the ratioof tail span to body length. For species with no marked tail (such as eels, leeches, or crocodiles), the maximum valueof the animal span in its posterior half have been taken in place of the tail span. To estimate the wetted surface of eachspecies, rough estimates have been used because this data was rarely given in the source papers.

As discussed above, measuring the drag on swimming fishes, and even defining it, is a difficult task and thedata found in the literature are not always consistent. An estimate is however needed if one wants to use Lighthill’selongated-body theory. A simple way to estimate the drag coefficient, which compares reasonably well with some ofthe available data (Lighthill, 1971; Videler, 1981; Webb et al., 1984; Fish and Rohr, 1999; Anderson et al., 2001), isto take the double of the drag coefficient for a flat plate in laminar flow for small Reynolds number, and the turbulentdrag coefficient of a flat plate for larger Reynolds number. The drag coefficient is then

Cd(Re) = max(2 × 1.328 Re−1/2, 0.072 Re−1/5

). (17)

The above relationship is usually an underestimate of the drag coefficient, particularly for animals with relatively poorstreamlining. The relation (17) has been used to calculate the drag coefficient for all animals except some mammals

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Species L (cm) S/h2 Re Li St U/V θ0 (deg)

MammalsBelugaa 364 7.3 8.0 × 106 0.145 0.35 0.48 31Bottlenose dolphina 258 7.9 1.2 × 107 0.063 0.26 0.52 25False killer whalea 379 7.8 2.1 × 107 0.044 0.26 0.57 28Florida manateeb 334 7.5 4.4 × 106 0.025 0.31 0.66Harp sealc 153 7.7 1.6 × 106 0.123 0.27 0.45 22Killer whalea 473 7.3 2.6 × 107 0.015 0.28 0.56 29Ringed sealc 106 8.7 1.3 × 106 0.105 0.30 0.44 24White-sided dolphina 221 6.5 1.3 × 107 0.018 0.24

SharksBlacktip reef sharkd 97 7.6 8.3 × 105 0.036 0.25 0.66Bonnethead sharkd 93 5.5 8.0 × 105 0.026 0.27 0.74Nurse shark d 220 17.8 1.8 × 106 0.072 0.41Scalloped hammerheade∗ 59 4.9 3.8 × 105 0.027 0.37

ScombridsAtlantic mackerel f 32 6.7 5.8 × 105 0.034 0.25 0.73Chub mackerelg 21 10.5 1.6 × 105 0.070 0.25Chub mackerelh∗ 21 10.5 1.8 × 105 0.067 0.26 0.63Giant bluefin tunai 250 4.8 5.7 × 106 0.015 0.24Kawakawa tunah∗ 21 5.0 1.8 × 105 0.032 0.21 0.60Pacific bonito j 47 6.2 4.5 × 105 0.033 0.23Skipjack tunak 57 5.8 2.2 × 106 0.022 0.27Yellowfin tunal 53 5.5 6.1 × 105 0.028 0.29 0.48

Table 1: Swimming kinematics for 53 different species of mammals, fish, amphibians and reptiles (continued in Tables 2,3). The different columnsare: the animal length, L; the surface ratio, S/h2; the Reynolds number, Re; the Lighthill number, Li; the Strouhal number, St, the slip ratio,U/V; and the maximum incident angle at the tail tip, θ0. The superscript ‘∗’ marks the juvenile animals and the superscripts < and > indicate theminimum and maximum values of the continuous dataset on the raibow trout (Webb et al., 1984). Sources: a: Fish (1998); Fish and Rohr (1999);Rohr and Fish (2004), b: Kojeszewski and Fish (2007), c: Fish et al. (1988), d: Webb and Keyes (1982), e: Lowe (1996), f : Videler and Hess(1984), g: Dickson et al. (2002), h: Donley and Dickson (2000), i: Wardle et al. (1989), j: Dowis et al. (2003), k: Yuen (1966), l: Dewar andGraham (1994).

(beluga, bottlenose dolphin, false killer whale, harp seals, killer whale and ringed seals) for which the authors provideda drag coefficient corrected for surface effects (Fish, 1998; Fish et al., 1988). For the different morphotypes of goldfishstudied by Blake et al. (2009), the drag coefficient has also been calculated from the data fit given by the authorsbecause these morphotypes have been selected artificially for aesthetic reasons, and as a result have a relatively largedrag.

From the literature, 89 different swimming kinematics have been identified. After analysis, 23 of these data havebeen discarded, either because better data were available for the same or a similar species, because the quality of thedata was doubtful (when it was based on a single experiment, for instance) or because the validity of elongated-bodytheory as defined by equation (16) was not ensured (this is mainly why there is no snake and no larva in the data set).The remaining 66 data represents 53 different species which have been divided in 7 different groups (Tables 1–3): 8different species of mammals, 4 of sharks, 8 of scombrids (a family which includes tunas, bonitos and mackerels),11 of fishes from the order of Perciformes and Salmoniformes (excluding the family of scombrids), 19 of fishesfrom other families (including Cypriniformes, Gadiformes and Mugiliformes), 10 of ‘elongated’ fishes (includingeels, needlefish of the family of Belonidae, and other fishes with surface ratio S/h2 greater than 17) and 6 speciescategorised as ’others’ gathering one reptile (crocodile), two frog tadpoles, two amphibians (axolotl and siren) andone annelid (leech).

8

Species L (cm) S/h2 Re Li St U/V θ0 (deg)

Perci/salmoni-formesAtlantic salmona 66 6.7 3.7 × 105 0.037 0.26 0.63Bluefishb 42 5.3 5.0 × 105 0.028 0.33Lake troutb 21 5.4 2.0 × 105 0.034 0.33Largemouth bassc 24.5 6.4 1.2 × 105 0.050 0.23 0.68 44Pacific jack mackereld 27 5.7 5.0 × 105 0.030 0.31Rainbow troute∗< 5.5 8.8 1.6 × 104 0.184 0.38 0.57Rainbow troute> 56 9.0 2.5 × 105 0.054 0.25 0.71Rainbow trout f 20.1 8.2 1.1 × 105 0.067 0.26 0.75 47Sockeye salmong 20.4 10.4 8.0 × 104 0.097 0.31 0.60Yellowbelly rockcodh∗ 7.6 22.2 2.3 × 104 0.385 0.38Yellowbelly rockcodh 29 10.5 2.2 × 105 0.065 0.30

Other fishesAtlantic codi 25 16.4 1.2 × 105 0.124 0.30 0.62Atlantic cod j 63 10.6 3.1 × 105 0.061 0.28Atlantic codk 49 10.6 3.7 × 105 0.059 0.25 0.75Atlantic silversidel 7.5 10.9 1.7 × 104 0.224 0.27Common breamm 19 3.4 8.5 × 104 0.031 0.29 0.76Common dace n 17.5 4.9 3.5 × 105 0.027 0.29Goldfish (Eggfish)o 5.3 10.6 7.1 × 103 0.538 0.54 0.41Goldfish (Fantail)o 5.7 10.4 7.7 × 103 0.512 0.47 0.53Goldfish (Common)o 5.1 4.7 2.1 × 104 0.093 0.40 0.80Goldfish (Comet)o 5.7 3.9 2.3 × 104 0.067 0.44 0.58Goldfishn 18.8 4.3 1.5 × 105 0.030 0.30Lake sturgeonp 15.7 12.8 4.1 × 104 0.168 0.48 0.65 46Mulletb 27 9.6 3.0 × 105 0.056 0.33Saitheq 36.4 7.0 3.9 × 105 0.038 0.23 0.76Thinlip grey mulleta 36 7.7 3.8 × 105 0.042 0.23 0.76Thicklip grey mulletr∗ 12.6 6.4 2.3 × 104 0.113 0.34 0.70 31Tiger musky f 18.3 9.4 9.6 × 104 0.081 0.25 0.60 50West African lungfishs 55 14.6 7.0 × 103 0.463 1.02West African lungfishs 55 14.6 6.0 × 104 0.158 0.75

Table 2: Same as Table 1. Sources: a: Videler (1993), b: Kayan et al. (1978), c: Jayne and Lauder (1995), d: Hunter and Zweifel (1971), e: Webbet al. (1984), f : Webb (1988), g: Webb (1973), h: Archer and Johnston (1989), i: Webb (2002), j: Webber et al. (2001), k: Videler (1993); Videlerand Wardle (1978), l: Parrish and Kroen (1988), m: Bainbridge (1963), n: Bainbridge (1958), o: Blake et al. (2009), p: Webb (1986), q: Hess andVideler (1984), r: Muller et al. (2002), s: Horner and Jayne (2008).

9

Species L (cm) S/h2 Re Li St U/V θ0 (deg)

ElongatedAtlantic needlefisha 23 21.7 1.2 × 105 0.167 0.34 0.69American eelb 21 25.8 6.0 × 104 0.280 0.31 0.73American eelc 36 25.8 1.3 × 105 0.192 0.37 0.79European eeld 22 27.9 4.0 × 104 0.369 0.48 0.60European eele 73 27.9 2.5 × 105 0.167 0.52Garfish f 44 18.0 4.0 × 105 0.098 0.34Great sand-eelg 30 19.2 1.2 × 105 0.148 0.31 0.67Hagfishh 31 33.2 6.4 × 104 0.347 0.56 0.49Lesser sand-eelg 9.0 16.4 2.2 × 104 0.296 0.41 0.64Longnose gari 57 21.9 3.3 × 105 0.125 0.59 0.67

OthersAxolotl j 17.7 19.2 4.4 × 104 0.242 0.57 0.59Bullfrog tadpolek∗ 4.7 11.0 2.3 × 104 0.193 0.79 0.58Green frog tadpolek∗ 5.0 10.6 2.1 × 104 0.195 0.60 0.60Lesser sirenl 34 31.3 1.7 × 105 0.202 0.54 0.61Medicinal leechm 10.0 19.5 1.8 × 104 0.384 0.63 0.70Saltwater crocodilen 93 62.5 4.2 × 105 0.339 0.78

Table 3: Same as Table 1. Sources: a: Liao (2002), b: Tytell and Lauder (2004), c: Gillis (1998), d: D’Aout and Aerts (1999), e: Ellerby et al.(2001), f : Kayan et al. (1978), g: Videler (1993), h: Long et al. (2002), i: Long et al. (1996), j: D’Aout and Aerts (1997), k: Wassersug and Hoff

(1985), l: Gillis (1997), m: Jordan (1998), n: Seebacher et al. (2003).

0.01 0.1 10.1

0.2

0.3

0.4

0.50.60.70.80.91.0

2.0

Li

St

MammalsSharksScombridsPerci/salmoni formesOther fishesElongatedOthersModel

Figure 4: Strouhal number of 53 different species of aquatic animals as a function of the Lighthill number. These animals are divided in differentcategories corresponding to the different symbols displayed in the legend. The solid line is the predicted optimal Strouhal number and the dashedline correspond to the interval for which efficiency is larger than ηmax − 0.1. The horizontal dotted lines correspond to the interval 0.25 < St < 0.35suggested by Triantafyllou et al. (1993). The solid line between two triangles corresponds to the continuous results of Webb et al. (1984) on therainbow trout.

10

0.01 0.1 10

10

20

30

40

50

60

70

80

90!

0

Li0.01 0.1 10

0.2

0.4

0.6

0.8

1

U/V

Li

(a) (b)

Figure 5: (a) Maximum angle at the tail tip θ0 and (b) slip ratio U/V as a function of the Lighthill number (same legend as in Fig. 4).

3.3. Results

The Strouhal number, the maximum angle at the tail tip and the slip ratio predicted by the present theoretical modelare compared to the observations on the different species in Figs. 4–5. In these figures, the thick line correspond tothe optimal case with Froude efficiency ηmax, and the dashed lines correspond to the ‘acceptable’ interval for whichthe Froude efficiency is η > ηmax − 0.1, as defined above (Fig. 3).

As seen in Fig. 4, the present analysis predicts that the optimal Strouhal number increases with the Lighthillnumber from 0.15 for the largest cetaceans to 0.8 for the smallest animals considered (or, more precisely, for animalswith the largest Lighthill number). This optimal Strouhal number curve can be approximated by a power law: St ≈0.75 Li1/3. Although the experimental observations are fairly scattered, this general trend is clearly observable for allthe aquatic animals, more than 85% of the data points having a Strouhal number within the acceptable range.

Among these data, the results of Webb et al. (1984) on the rainbow trout stand out. They studied animals with totallength ranging from L = 5.5 cm to 56 cm and deduced from hundreds of measurements how the different geometricand kinematic quantities varies with the length and the swimming speed of the trouts. This allows, for a single species,to see how the Strouhal number varies with the Lighthill number (the thin solid line in Fig. 4). Remarkably, this lineis parallel (on the log-log scale of Fig. 4) to the the theoretical prediction.

Because equation (17) probably tends to underestimate the drag coefficient, the actual Lighthill number may belarger than the estimates used in Figs. 4–5. For animals with relatively poor streamlining, this increase may be aslarge as a factor 2 or 3. It means that error bars on Li are rather large and that some data points may have to be shiftedto larger values of Li. As for the Strouhal number, the typical range of variation between different animals or betweendifferent measurements on the same animal is generally of the order of ±25% which would give a conservative estimateof the error bars on St. It can be argued however that the optimal swimming cases probably correspond to the smallestobserved St and thus some data points of Fig. 4 may have to be shifted to smaller values of St.

The comparison between the predicted maximum angle at the tail tip and the experimental observations (Fig. 5a)is less conclusive, mostly because of the lack of data and because the acceptable range is fairly large. However, theobservations for mammals (θ0 ≈ 25 deg) and fishes (θ0 ≈ 45 deg) fall within the predicted range.

The predicted slip ratio U/V has also been compared with observations on animals (Fig. 5b). In each group, theslip ratio is decreasing with the Lighthill number, as predicted, but the mammals and the scombrids are clearly belowthe prediction, while elongated fishes are clearly above. This discrepancy will be discussed below. Another featureof the slip ratio is that the optimal case corresponds to a maximum: for a given Lighthill number, when efficiency islower than the optimal, so is U/V .

Note that, again, the slip ratio deduced from the results of Webb et al. (1984) on the rainbow trout (the thin solidline in Fig. 5b) agrees remarkably well with the present prediction.

11

0.01 0.1 10.1

0.2

0.3

0.4

0.50.60.70.80.91.0

2.0

Li

St

MammalsSharksScombridsPerci/salmoni formesOther fishesElongatedOthersModel

103 104 105 106 107 1080

0.2

0.4

0.6

0.8

1

St

Re

0.01 0.1 10.1

0.2

0.3

0.4

0.50.60.70.80.91.0

2.0

Li

St

MammalsSharksScombridsPerci/salmoni formesOther fishesElongatedOthersModel

Figure 6: Strouhal number as a function of the Reynolds number (same legend as in Fig. 4).

4. Discussion

In this paper, Lighthill’s elongated-body theory has been used to predict the optimal Strouhal number for swim-ming animals. Using the elongated-body assumptions, it appeared that the optimal Strouhal number depends on asingle dimensionless quantity, which has been called the Lighthill number, and which can be regarded as the ratioof the animal drag to its typical achievable thrust. Together with the optimal Strouhal number, were predicted themaximum incident angle at the tail tip and the slip ratio, which also depend uniquely on the Lighthill number. Thesetheoretical predictions for optimal motion of the tail tip have been then compared with the swimming kinematics of53 different species of swimming animals. It appeared that the general trends predicted by the present model arerecovered in the zoological data, indicating that animals generally swim near the predicted optimum (Figs. 4–5).

Additionally, the present model elucidates three previous unexplained observations: first, the Strouhal number,St, was shown to be a decreasing function of the animal velocity (Fig. 11.3B in Lauder and Tytell, 2005); then, Stwas observed to decrease as an animal grows (Webb et al., 1984) and finally, St has been measured to decrease withincreasing Reynolds number as different species are considered (Kayan et al., 1978).

In Fig. 6, the Strouhal number has been plotted as a function of the Reynolds number for the same species. Ageneral trend can be identified, showing that St is generally decreasing with Re. This is compatible with the presentmodel, although the aspect ratio of the animal is not taken into account when considering the dependence on Reonly. Moreover, there is no physical argument that could explain this dependence on the Reynolds number. Based onanalogies with bluff bodies wakes, one could even expect the opposite trend (i.e. St increasing with Re).

The validity of the elongated-body theory is limited by two geometric quantities. First, the variations of the cross-section should occur on typical scales of the order of the animal length. This is clearly not the case for animals withhigh aspect-ratio tails (also called lunate tails) like cetaceans, scombrids and sharks. For sharks, additional difficultyis caused by the asymmetry of the tail and one could ask whether the Strouhal number should be based on the motionof the largest lobe, the smallest lobe, or some average of the two. Second, the elongated-body theory is not adaptedto anguilliform animals like eels for which the tail depth is difficult to define. For these very elongated animals,an additional effect come into play as resistive forces cannot be neglected anymore, as pointed out above. Theselimitations probably explain why the slip ratio for the cetaceans and the scombrids is approximately 0.2 smaller thanpredicted while the elongated fishes seem to have a slip ratio larger than predicted.

As noted above, the key feature of Lighthill’s elongated-body theory is that the two quantities needed to performthe optimisation, namely the average propulsive thrust and the average power loss in the wake, only depend on localquantities evaluated at the tail tip. This property has been essential in developing the present model, but a naturalquestion would be now to ask whether the predicted optimal tail motions are compatible with the complete kinematicsof a swimming animal. In particular, it would be important to evaluate the role of recoil (Webb, 1992), the effect ofpassive elasticity of the tail and the role of the internal mechanics in general (see Cheng et al., 1998; Long, 1992;

12

(a)

L

h

(a) (b)

Figure 7: Schematic three-dimensional views of the (a) BvK and (b) rBvK vortex streets, corresponding to the two-dimensional views of Fig. 1.

Long et al., 2002; McMillen and Holmes, 2006, for instance).Let us now examine the relation between the Strouhal number and the characteristics of the wake. First, it may

be important to remind that Lighthill’s elongated-body theory includes a wake behind the swimmer (see Fig. 7 inCandelier et al., 2011, for instance). This wake is composed of an infinitely thin sheet of vorticity left by the passageof the trailing edge in the water. Applying Kelvin’s circulation theorem, this wake is found to be composed of flattenvortex rings and contains the kinetic energy given by the animal to the fluid. But, in Lighthill’s elongated-body theory,the dynamical evolution of this wake is not described because it has no influence on the dynamics of the swimmer.

One plausible scenario for the wake is that the vorticity sheet predicted by Lighthill’s elongated-body theorywill eventually roll-up to form a chain of vortex rings as sketched in Fig. 7b. The wake would then resemble theexperimental observations of different groups (Blickhan et al., 1992; Lauder and Tytell, 2005; Muller et al., 1997;Nauen and Lauder, 2002; Videler, 1993). Another possible scenario is that, due to their self-induced velocities, theseconcatenated vortex rings will separate in two rows of vortex rings as observed for eels both experimentally (Lauderand Tytell, 2005; Muller et al., 2001; Tytell and Lauder, 2004) and numerically (Borazjani and Sotiropoulos, 2008;Kern and Koumoutsakos, 2006). In both cases, each vortex rings is expected to have a vertical extension approximatelyequal to the tail span and a horizontal extension approximately equal to U/2 f . This means that the aspect ratio of thevortex rings varies from near circular to elongated in the swimming direction (for elongated fishes). As a rule, there is amajor difference between the inherently three-dimensional wake behind a swimming animal and the two-dimensionalwake observed behind an infinite cylinder (as drawn in Fig. 7).

Now coming back to the results of Triantafyllou et al. (1991, 1993) on the stability and efficiency of wakes, itappears that both their theoretical model and their experiments (Anderson et al., 1998; Read et al., 2003; Schouveileret al., 2005; Triantafyllou et al., 1993) are based on two-dimensional flows. If this limit case can be suited to animalswith large aspect-ratio tails, some specific work seems needed to apply it to other animals.

For animals with large aspect-ratio tails though, one striking fact is that the present model (where the wake hasno influence) and the theoretical predictions of Triantafyllou et al. (1991, 1993) (based on the wake characteristics)converge to give a similar interval for the Strouhal number: roughly 0.2 < St < 0.4. One possibility is that theseanimals have evolved to optimise both the formation of a coherent wake and the elongated-body efficiency. Thiswould explain why the geometrical characteristics of these animals, in particular the ratio of tail span to body length,vary so little among the species.

Acknowledgment

The author acknowledges support from the European Commission through a Marie Curie IOF fellowship (PIOF-GA-2009-252542).

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